Effects of nonlinear strain components on the buckling response of stiffened shear-deformable composite plates

A detailed investigation of the weight of each non linear term of the Green-Lagrange strain displacement equation is presented, with reference to the buckling of orthotropic, both flat and prismatic, Mindlin plates. Usually in the literature, in buckling analysis only the second order terms related to the out-of-plane displacement are considered. Such heuristic simplification, known as von Kármán hypothesis, starts by the consideration that the buckling mode of a flat plate is described by dominant out-of-plane displacement and disregards the non-linear terms of the Green-Lagrange strain tensor depending on the in plane displacement components, whose role is confined to first order, say pre-critical, deformation. The present paper shows that disregarding the non linear terms related to the in-plane strain-displacement is equivalent to neglect shear induced rotation. In the work, the governing equations are derived using the principle of strain energy minimum and the differential equations solution is gained by using the general Levy-type method. The obtained results show that the von Kármán model overestimates the critical load when, in buckling mode, magnitudes of shear rotation, in-plane and out-of-plane displacements are comparable.